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Estimation of Saturation Exponent from NuclearMagnetic Resonance (NMR) Logs in LowPermeability Reservoirs
Liang Xiao • Zhi-qiang Mao • Gao-ren Li • Yan Jin
Received: 26 November 2011 / Revised: 3 May 2012 / Published online: 3 June 2012
� Springer-Verlag 2012
Abstract The resistivity experimental measurements of 36 core samples, which
were drilled from low permeability reservoirs of southwest China, illustrate that the
saturation exponents are not agminate, but vary from 1.627 to 3.48; this leads to a
challenge for water saturation estimation in low permeability formations. Based on the
analysis of resistivity experiments, laboratory nuclear magnetic resonance (NMR)
measurements for all 36 core samples, and mercury injection measurements for 20 of
them, it was observed that the saturation exponent is proportional to the proportion of
small pore components and inversely proportional to the logarithmic mean of NMR T2
spectrum (T2lm). For rocks with high proportion of small pore components and low
T2lm, there will be high saturation exponents, and vice versa. The proportion of small
pore components is characterized by three different kinds of irreducible water satu-
rations, which are estimated by defining 30, 40 and 50 ms as T2 cutoffs separately. By
integrating these three different kinds of irreducible water saturations and using T2lm, a
technique of calculating the saturation exponent from NMR logs is proposed and the
corresponding model is established. The credibility of this technique is confirmed by
L. Xiao (&)
Key Laboratory of Geo-detection, China University of Geosciences, Beijing, Ministry of Education,
No. 29, Xueyuan Road, Haidian, Beijing 100083, People’s Republic of China
e-mail: [email protected]
Z. Mao
College of Geophysics and Information Engineering, China University of Petroleum,
Beijing, People’s Republic of China
G. Li
Research Institute of Exploration and Development, Changqing Oilfield Company,
PetroChina, Shaanxi, People’s Republic of China
Y. Jin
Southwest Oil and Gas Field Branch Company, PetroChina,
Sichuan, People’s Republic of China
123
Appl Magn Reson (2013) 44:333–347
DOI 10.1007/s00723-012-0366-1
Applied
Magnetic Resonance
comparing the predicted saturation exponents with the results from the core analysis.
For more than 85 % of core samples, the absolute errors between the predicted satu-
ration exponents from NMR logs and the experimental results are lower than 0.25.
Once this technique is extended to field application, the accuracy of water saturation
estimation in low permeability reservoirs will be improved significantly.
1 Introduction
Water saturation (thus related to hydrocarbon saturation) is an indispensable input
parameter in formation evaluation, and it also plays a very important role in
reservoir development program formulation. Generally, water saturation is calcu-
lated using Archie’s equations after the necessary parameters have been obtained
[1]. Archie’s equations can be expressed as Eqs. (1) and (2):
F ¼ R0
Rw
¼ a
/m ð1Þ
Ir ¼Rt
Ro
¼ b
Snw
ð2Þ
where R0 is the rock resistivity at full water saturation, Rt the true formation
resistivity, Rw the formation water resistivity, the units of which are X m, F the
formation factor, Ir the resistivity index, / the porosity in fraction, a and b the
lithology factors, m the cementation exponent, Sw the water saturation in fraction,
and n is the saturation exponent.
Combining with Eqs. (1) and (2), a derivative expression can be written as follows:
Sw ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a� b� Rw
/m � Rt
n
s
: ð3Þ
From Eq. (3), it can be observed that the values of a, b, m, n, Rw, / and Rt must
be obtained first for the water saturation calculation, / and Rt can be acquired from
conventional logs [2–4], and Rw can be checked from the formation water salinity
using Schlumberger’s log interpretation charts [5].
2 Determination of the Values of a, b, m and n
To calculate water saturation from conventional logs using Archie’s equation, the
determinations of the values of a, b, m and n are crucial. Generally, the
determinations of a, b, m and n rely on the resistivity experimental measurements of
the target core samples. To obtain the necessary resistivity experimental data, the
needed procedures should be applied as follows: (1) every waterless core sample is
saturated using the used saline water, and the rock resistivity R0 at full water
saturation is measured; in this study, the salinity of the used saline water is
13.00 mg/l. (2) The oil is used as the displacing medium, and the centrifugal method
is used to vary the water saturation (Sw) of core samples, and the corresponding rock
334 L. Xiao et al.
123
resistivity Rt of every core sample under different water saturations are measured.
(3) R0, Rt and Sw are collected as a data set to obtained the value a, b, m and n.
For conventional reservoirs, after the representative core samples were drilled
from the intended intervals for the resistivity experiment, the fixed values of a, b,
m and n can separately be obtained from the cross plots of the porosity with the
formation factor, and the water saturation with the resistivity index using the power
function. However, for low permeability sands, not rigorous power function exists
between the porosity and the formation factor, the water saturation and the resistivity
index due to the complicated pore structure and the strong heterogeneity [6]. Wang
and Sharma [7] and Mao et al. [8, 9] had proposed that the tendency of porosity and
formation factor would be changed when the porosities of core samples are lower
than 9.0 %, and they had demonstrated that this change is caused by the poor pore
structure of low permeability plug samples. Mao et al. [8] had developed a novel
method to obtain the accurate values of a and m from the porosity using binary
regression. This method has been confirmed to be effective and is used widely [6]. In
the low permeability formations mentioned in this study, a and m can be determined
using Mao’s method precisely. Thus, the technique of determining a and m from the
porosity that has been proposed by Mao et al. [8] is not introduced in this paper.
It is really a challenge to determine the saturation exponent in low permeability
reservoirs, as the cross plot of the water saturation with the resistivity index is
divergent and a fixed saturation exponent is difficult to acquire. Figure 1 shows the
cross plot of the water saturation with the resistivity index of 36 core samples,
which were drilled from low permeability reservoirs of southwest China. It can be
observed that the relationship between the water saturation and the resistivity index
for all core samples is not consistent. The saturation exponent for single core sample
varies from 1.627 to 3.48. In this case, water saturation calculated using the
regressed fixed saturation exponent from all 36 core samples would be inaccurate.
y = 1.0415x-2.0797
R2 = 0.9054
1
10
0.1 1
Water saturation, fraction
Res
istiv
ity e
xpon
ent,
Ir
Fig. 1 Relationship of water saturation and resistivity index for 36 core samples in low permeabilitysands of southwest China
Estimation of Saturation Exponent from NMR Logs 335
123
The best method is to estimate the water saturation using various saturation
exponents along with the target intervals.
3 Influencing Factors of Saturation Exponent in Low Permeability Sandstones
To acquire accurate saturation exponents for water saturation estimation at low
permeability, it is necessary to understand the influencing factors and the variation
of the saturation exponent. Based on the qualitative analysis of the core thin section
and mercury injection capillary pressure experimental data, Mao et al. [8] had point
out that the saturation exponents were related to rock pore structure. However, the
quantitative relationship between them was not established, and an applicable
technique was not proposed. Nuclear magnetic resonance (NMR) logs have a unique
advantage in indicating reservoir pore structure. From the measured NMR T2
distribution, the information of pore size and distribution can be obtained [10–14].
Rocks with macropore and good pore structure will display long transversal
relaxation time, and wide T2 distribution due to the contribution of surface
relaxation. On the contrary, short transversal relaxation time and narrow NMR T2
distribution mean poor pore structure for rocks (Fig. 2). Mercury injection capillary
pressure curves can be used to obtain the pore throat radius distribution, which is
useful in evaluating the pore throat size and the connectivity [15, 16].
To quantitatively display the relationship between the saturation exponents with the
pore structure, all 36 core samples, shown in Fig. 1, have been chosen for rock
resistivity and laboratory NMR experimental measurements; 20 of them were studied
in mercury injection experiments. The experimental parameters of NMR measure-
ments are designed as follows: polarization time (TW): 6.0 s; inter-echo spacing (TE):
0.2 ms; the number of echoes per echo train (NE): 4096; scanning number: 128.
To illustrate the factors that heavily affect the saturation exponent, the resistivity
and laboratory NMR experimental results for 36 core samples and mercury injection
measurements for 20 core samples have been analyzed. Four representative core
samples with saturation exponent increasing from 1.627 to 3.48 are compared and
displayed through Figs. 3, 4, 5 and 6. In these figures, the correlation of the water
saturation and the resistivity index, the corresponding laboratory NMR T2
distribution and the pore throat radius distribution that acquired from mercury
injection capillary pressure curve are displayed in (a), (b) and (c), respectively. For
the core sample no. 2, no mercury injection data have been obtained.
From a comparison of the data displayed in Figs. 3, 4, 5 and 6, several
regularities can be observed:
1. The saturation exponent is heavily affected by the rock pore structure. For core
samples with good pore structure (with wide NMR T2 distribution), the
proportion of large pore components is dominated and the corresponding
saturation exponent is low, like for the core samples 1 and 2 shown in Figs. 3
and 4. On the other hand, when the rocks are dominated by micro porosity, the
proportion of small pore components is high, and the corresponding saturation
exponent increase, like for the core samples 3 and 4 shown in Figs. 5 and 6.
336 L. Xiao et al.
123
2. The saturation exponent is hardly affected by the pore throat radius, as for the
core samples 1, 3, and 4. Their saturation exponents vary strongly, but their
distributions of pore throat radii are not different, especially for the core
samples 1 and 3. Their saturation exponents and NMR T2 distributions are
significantly different, while the morphologies of the pore throat radius
distributions are almost the same.
3. The saturation exponent is not relevant to rock porosity and permeability, but it
is inversely proportional to T2lm. This is because high T2lm means wide NMR T2
distributions and thus leads to low saturation exponents.
4 A Novel Model for Estimating the Saturation Exponent from NMR Logs
4.1 Estimating the Saturation Exponent Parameters from NMR Logs
From Figs. 3, 4, 5 and 6, we can conclude that the saturation exponent is
proportional to the proportion of small pore components and inversely proportional
0.1 1 10 100 1000
T 2
T2
-
0.1 1 10 100 1000
Rel
ativ
e am
plitu
de
Relaxation time , ms T 2Relaxation time , ms
Rel
ativ
e am
plitu
de
H1 nuclei
Equivalent rock pore space
Spin-echo train
NMR distribution
Multi- exponential
inversion
0 300 600 900
Decay time, ms
Am
plitu
de
0 300 600 900
Decay time, ms
Am
plitu
de
Fig. 2 Relationship of rock pore size with the corresponding NMR T2 distribution
Estimation of Saturation Exponent from NMR Logs 337
123
to the T2lm. These parameters must be obtained first to estimate the saturation
exponent precisely. In this aspect, NMR logs have unique advantages [17–21]. T2lm
can be obtained from the NMR logs directly, but the proportion of small pore
components needs to be characterized.
In this study, different kinds of irreducible water saturations, which are calculated
by defining 10, 20, 30, 40, 50, 60, 70 and 100 ms as T2 cutoffs separately, are
y = 1.0252x-1.627
R2 = 0.998
1
10(a)
(b)
(c)
0.1 1
Water saturation, fraction
Res
istiv
ity in
dex
Core No. 1
por.=14.0%
perm.=0.49 mD
0
0.05
0.1
0.15
0.2
0.25
0.3
0.1 10 1000 100000
T 2, ms
Am
plitu
de
Core No. 1
T 2lm =30.6 ms
0
10
20
30
0.01 1 100 10000
R c, um
Am
plitu
de
Core No. 1
Fig. 3 Experimental results ofcore sample no. 1
338 L. Xiao et al.
123
chosen to characterize the proportion of small pore components. The irreducible
water saturation can be estimated using Eq. (4),
Swirr ¼R T2cutoff
T2 minSðTÞdt
R T2 max
T2 minSðTÞdt
ð4Þ
where Swirr is the estimated irreducible water saturation from NMR logs using the
defined T2 cutoff, T2min the minimum transverse relaxation time, T2max the
maximum transverse relaxation time, T2cutoff the defined T2 cutoff, which is used to
estimate the irreducible water saturation; the units of them are ms and S(T) is the
porosity distribution function, which is associated with the T2 relaxation time.
To illustrate the correlation of all the experimental parameters obtained from
laboratory NMR and mercury injection measurements with the saturation exponent,
the correlations of them are analyzed and listed in Table 1.
Table 1 illustrates that the saturation exponents are strongly correlated with
Swirr_30, Swirr_40, Swirr_50, but the correlation with Swirr_10, Swirr_20, Swirr_60, Swirr_70
and Swirr_100 was reduced. This is because that for majority of core samples, the
y = 1.0132x-1.885
R2 = 0.9982
1
10(a)
(b)
0.1 1
Water saturation, fractionR
esis
tivity
inde
x
Core No. 2
por.=15.9%
perm.=1.05 mD
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.1 10 1000 100000
T 2, ms
Am
plitu
de
Core No. 2
T 2lm=75.182 ms
Fig. 4 Experimental results ofcore sample no. 2
Estimation of Saturation Exponent from NMR Logs 339
123
NMR T2 distribution mainly ranges from 20 to 60 ms. When the T2 relaxation time
is lower than 30 ms and higher than 60 ms, nearly no T2 spectrum exists. T2lm is the
overall signature of NMR T2. Hence, it is associated with the pore structure.
y = 0.9828x-2.234
R2 = 0.9955
1
10(a)
(b)
(c)
0.1 1
Water saturation, fractionR
esis
tivity
inde
x
Core No. 3
por.=8.2%
perm.=0.27 mD
0
0.05
0.1
0.15
0.2
0.1 10 1000 100000
T 2, ms
Am
plitu
de Core No. 3
T 2lm =28.2 ms
0
10
20
0.01 1 100 10000
R c, um
Am
plitu
de
Core No. 3
Fig. 5 Experimental results ofcore sample no. 3
340 L. Xiao et al.
123
Core porosity, permeability, T2 cutoff and parameters obtained from the
mercury injection measurements are weakly correlated with the saturation
exponent.
y = 0.9609x-3.4801
R2 = 0.9969
1
10(a)
(b)
(c)
0.1 1
Water saturation, fraction
Res
istiv
ity in
dex
Core No. 4
por.=10.58%
perm.=0.62 mD
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.1 10 1000 100000
T 2, ms
Am
plitu
de
Core No. 4
T 2lm =20.41 ms
0
10
20
0.01 1 100 10000
R c, um
Am
plitu
de
Core No. 4
Fig. 6 Experimental results ofcore sample no. 4
Estimation of Saturation Exponent from NMR Logs 341
123
Ta
ble
1C
orr
elat
ion
so
fth
esa
tura
tion
exp
onen
tan
dth
eex
per
imen
tal
par
amet
ers
ob
tain
edfr
om
lab
ora
tory
NM
Ran
dm
ercu
ryin
ject
ion
mea
sure
men
ts
Sat
ura
tio
nex
po
nen
tP
oro
sity
Per
mea
bil
ity
T2cuto
fflo
g(T
2lm
)S
wir
r_10
Sw
irr_
20
Sw
irr_
30
Sw
irr_
40
Sw
irr_
50
Sat
ura
tio
nex
po
nen
t1
.00
Po
rosi
ty-
0.4
91
.00
Per
mea
bil
ity
-0
.27
0.4
41
.00
T2cuto
ff-
0.4
80
.58
0.6
11
.00
log
(T2lm
)-
0.5
80
.66
0.4
90
.67
1.0
0
Sw
irr_
10
0.5
2-
0.6
5-
0.3
2-
0.6
2-
0.9
61
.00
Sw
irr_
20
0.6
1-
0.6
6-
0.4
2-
0.6
6-
0.9
80
.97
1.0
0
Sw
irr_
30
0.6
5-
0.6
6-
0.4
9-
0.6
4-
0.9
70
.92
0.9
81
.00
Sw
irr_
40
0.6
5-
0.6
5-
0.5
2-
0.6
1-
0.9
60
.87
0.9
50
.99
1.0
0
Sw
irr_
50
0.6
4-
0.6
4-
0.5
4-
0.5
8-
0.9
50
.84
0.9
20
.98
1.0
01
.00
Sw
irr_
60
0.6
2-
0.6
2-
0.5
6-
0.5
7-
0.9
40
.81
0.9
00
.96
0.9
91
.00
Sw
irr_
70
0.6
1-
0.6
0-
0.5
7-
0.5
7-
0.9
30
.80
0.8
80
.95
0.9
80
.99
Sw
irr_
100
0.5
6-
0.5
5-
0.6
1-
0.5
9-
0.9
10
.76
0.8
40
.91
0.9
40
.96
So
rtin
gco
effi
cien
t-
0.1
50
.64
0.4
40
.40
0.6
6-
0.6
1-
0.6
0-
0.6
3-
0.6
5-
0.6
7
Var
iati
on
coef
fici
ent
-0
.30
0.7
30
.79
0.6
20
.76
-0
.64
-0
.69
-0
.74
-0
.78
-0
.80
P50
0.5
1-
0.7
1-
0.2
9-
0.5
0-
0.7
80
.78
0.7
70
.78
0.7
80
.77
R50
-0
.29
0.5
51
.00
0.7
20
.60
-0
.42
-0
.51
-0
.58
-0
.62
-0
.64
Pd
0.1
9-
0.5
8-
0.2
5-
0.4
1-
0.6
60
.64
0.6
00
.60
0.6
00
.61
Rm
ax
-0
.33
0.6
90
.89
0.6
60
.70
-0
.54
-0
.63
-0
.70
-0
.75
-0
.77
Rm
-0
.31
0.6
10
.99
0.7
10
.65
-0
.4-
0.5
7-
0.6
4-
0.6
8-
0.7
1
342 L. Xiao et al.
123
Ta
ble
1co
nti
nu
ed
Sw
irr_
60
Sw
irr_
70
Sw
irr_
100
So
rtin
gco
effi
cien
tV
aria
tio
nco
effi
cien
tP
50
R50
Pd
Rm
ax
Rm
Sat
ura
tio
nex
po
nen
t
Po
rosi
ty
Per
mea
bil
ity
T2cuto
ff
log
(T2lm
)
Sw
irr_
10
Sw
irr_
20
Sw
irr_
30
Sw
irr_
40
Sw
irr_
50
Sw
irr_
60
1.0
0
Sw
irr_
70
1.0
01
.00
Sw
irr_
100
0.9
80
.99
1.0
0
So
rtin
gco
effi
cien
t-
0.6
8-
0.6
8-
0.6
71
.00
Var
iati
on
coef
fici
ent
-0
.82
-0
.83
-0
.85
0.8
91
.00
P50
0.7
60
.75
0.7
0-
0.7
4-
0.6
91
.00
R50
-0
.67
-0
.68
-0
.75
0.4
50
.80
-0
.32
1.0
0
Pd
0.6
10
.61
0.5
9-
0.8
9-
0.7
10
.83
-0
.27
1.0
0
Rm
ax
-0
.80
-0
.81
-0
.84
0.7
40
.95
-0
.52
0.8
9-
0.5
11
.00
Rm
-0
.73
-0
.74
-0
.80
0.5
70
.88
-0
.40
0.9
9-
0.3
60
.95
1.0
0
Inth
ista
ble
,S
wir
r_10,
Sw
irr_
20,
Sw
irr_
30,
Sw
irr_
40,
Sw
irr_
50
,S
wir
r_60,
Sw
irr_
70
and
Sw
irr_
100
are
the
irre
du
cib
lew
ater
satu
rati
on
sca
lcu
late
dfr
om
the
NM
RT
2d
istr
ibu
tio
nu
sin
g
10
,2
0,
30
,4
0,
50
,6
0,
70
and
10
0m
sas
T2
cuto
ffs
P50
isth
em
ercu
ryin
ject
ion
pre
ssu
reco
rres
po
ndin
gto
50
.0%
mer
cury
inje
ctio
nsa
tura
tio
n,
R50
isth
ep
ore
thro
atra
diu
sco
rres
po
nd
ing
to5
0.0
%m
ercu
ryin
ject
ion
satu
rati
on
,P
dis
the
thre
sho
ldp
ress
ure
,R
max
isth
em
axim
um
po
reth
roat
rad
ius,
Rm
isth
eav
erag
ep
ore
thro
atra
diu
s
Estimation of Saturation Exponent from NMR Logs 343
123
4.2 A Novel Model of Estimating Saturation Exponent from NMR Logs
Based on the analysis described above, Swirr_30, Swirr_40, Swirr_50 and T2lm are chosen
as the input parameters to establish a model to estimate the saturation exponent.
With the 36 studied core samples, multivariate regression is used. The regression
model is established and expressed as Eq. (5).
n ¼ 0:546þ 0:292� logðT2lmÞ þ 0:009� Swirr 30 þ 0:061� Swirr 40
� 0:044� Swirr 50; correlation coefficient: 0:776ð5Þ
Equation (5) illustrates that the precision of the saturation exponent estimation
model is improved when the parameters Swirr_30, Swirr_40, Swirr_50 and T2lm are
introduced. If these three fixed T2 cutoffs of 30, 40 and 50 ms are determined, the
proportions of small pore components could be characterized and the consecutive
saturation exponents can be estimated from the NMR field logs after this technique
is extended to field application.
0 1 2 3 4 5
Saturation exponent
Measured n
Predicted n
Fig. 7 Comparison ofsaturation exponents acquiredfrom experimental resistivitymeasurements of core samplesand calculated from NMR fieldlogs
344 L. Xiao et al.
123
5 Case Studies
To confirm the reliability of the mentioned technique in this study, saturation
exponents acquired from the experimental resistivity measurements of core samples
and calculated from NMR field logs are compared in Fig. 7. This comparison shows
that for the vast majority of core samples, the predicted saturation exponents are
close to the experimental results. To quantitatively evaluate the absolute errors of
the predicted saturation exponents and the core results, the cross plot of these two
kinds of saturation exponents is made and shown in Fig. 8. These two figures
illustrate that the estimated saturation exponents from NMR field logs using the
proposed technique are credible and the absolute errors for more than 85 % of core
samples are lower than 0.25. In reservoirs with consecutive NMR field logs, this
technique can be applied for saturation exponent estimation and this will be
valuable for water saturation calculation in low permeability sands.
6 Conclusions
In low permeability reservoirs, the saturation exponents are divergent and a fixed
value cannot be regressed from the cross plot of the water saturation with the
resistivity index to estimate water saturation accurately.
1
2
3
4
1 2 3 4
Experimental saturation exponents from core samples
Pred
icte
d sa
tura
tion
expo
nent
s fr
om N
MR
fie
ld lo
gs
+0.25
-0.25
Fig. 8 Cross plot of the predicted saturation exponents and the core results
Estimation of Saturation Exponent from NMR Logs 345
123
The rock resistivity, laboratory NMR and mercury injection measurements of
core samples illustrate that the saturation exponent is heavily affected by the rock
pore structure. Thus, it is proportional to the proportion of small pore components
and inversely proportional to T2lm. The saturation exponent is not relevant to rock
porosity, permeability and the rock pore throat radius distribution.
The proportion of small pore components can be characterized by the irreducible
water saturations predicted from the NMR T2 distribution after defining 30, 40 and
50 ms as the fixed T2 cutoffs. An estimation model for the saturation exponent can
be established based on the corresponding irreducible water saturations and T2lm.
The saturation exponents predicted from NMR field logs using the proposed
model in this paper are credible, and they are close to the measured core results. For
more than 85 % of the core samples, the absolute errors of these two kinds of
saturation exponents are lower than 0.25. This ensures that the proposed technique
and model are reliable and can be extended to field application to estimate the
saturation exponents from NMR field logs. This is valuable for water saturation
calculation in low permeability sandstones.
Acknowledgments The authors thanks for the supporting of the Fundamental Research Funds for the
Central Universities, China (No. 2011YXL009) to this research work.
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